Note: In the following the typographical distinction
between vectors and scalars is that a vector is shown in red.
This distinction
has a visual impact but the nature of a variable is usually
readily
apparent from the context in which it is used.

Consider two coordinate systems, one a fixed (inertial) rectangular system
with unit vectors I, J and K, the other rotating on the surface of a sphere of
radius R. The latter has unit vectors i, j, k. The sphere is rotating
at a constant
rate of Ω. In the inertial coordinate system the rotation vector is
ΩK. In the
rotating coordinate system, where the unit vector
i
is to the east, the unit
vector j is to the north and the
unit vector
k is vertically normal to the
surface, the rotation vector Ω has the representation

Ω = Ωcos(φ)j
+ Ωsin(φ)k

where φ is the latitude angle.

An arbitrary vector A has
representations in
both coordinate systems; i.e.,

A = AxI
+ AyJ
+ AzKA = A'xi
+ A'yj +
A'zk

When considering the change in the vector A with time the crucial difference
between the two coordinate systems is that in the inertial system the unit
vectors are constant whereas in the rotating system they are not. Thus,

(dA'x/dt)i + (dA'y/dt)j + (dA'z/dt)k

represent the apparent time rate of change of A in the rotating coordinate
system, which can be denoted as drA/dt.
Thus

dA/dt = drAdt)
+ A'xdi/dt + A'ydj/dt +
A'zdk/dt

The Time Derivatives of the Tangent Plane Unit Vectors of a Rotating
Coordinate System

The point of origin of the tangent plane coordinate system can be
expressed in terms of the inertial frame either in rectangular coordinates
(x, y, z) or spherical coordinates (r, θ, φ), where r is
the
radius of the sphere, θ is the longitude and φ is the latitude.
The relationship between the two representations is:

x = rcos(φ)cos(θ)
y = rcos(φ)sin(θ)
z = rcos(φ)

The local vertical unit vector k at (x, y, z)
in terms of the inertial
frame is:

dj/dt = Ωxj.

The General Form of the Time Derivative of a Vector

dA/dt = drA/dt
+ A'xdi/dt + A'ydj/dt +
A'zdk/dt

If the time derivatives of the local unit vectors, di/dt, dj/dt and dk/dt,
are replaced by their values as Ωxi, Ωxj and Ωxk,
and the rotation vector Ω factored from the cross products the result
is:

dA/dt = drA/dt
+ Ωx(A'xi + A'yj +
A'zk)

which is none other than

dA/dt = drA/dt + ΩxA

This says that the time derivative of a vector can be constructed from
its apparent time derivative in the rotating frame plus the vector which
is the vector cross product of the rotation vector for the frame and the
vector itself. There are number of places in the literature where the
time derivatives of the unit basis vectors are derived from the above
formula on the basis of the argument that such unit vectors are just
special cases of position vectors to which the formula applies. This
is in valid because the formula has to be derived from the determination
of the time derivatives of those basis vectors. The formula does of
course apply to the basis vectors but it is logically invalid to
derive its application to the basis vectors from the formula itself.

The derivation strictly holds for position vectors and
its extension to axial vectors (vectors such as angular momentum and torque
which are vector cross products of position vectors) requires additional
analysis.
For this extension see Axial Vectors.